The global-in-time existence of bounded weak solutions to a large class of physically relevant, strongly coupled parabolic systems exhibiting a formal gradient-flow structure is proved. The main feature of these systems is that the diffusion matrix may be generally neither symmetric nor positive semi-definite. The key idea is to employ a transformation of variables, determined by the entropy density, which is defined by the gradient-flow formulation. The transformation yields at the same time a positive semi-definite diffusion matrix, suitable gradient estimates as well as lower and/or upper bounds of the solutions. These bounds are a consequence of the transformation of variables and are obtained without the use of a maximum principle. Several classes of cross-diffusion systems are identified which can be solved by this technique. The systems are formally derived from continuous-time random walks on a lattice modeling, for instance, the motion of ions, cells, or fluid particles. The key conditions for this approach are identified and previous results in the literature are unified and generalized. New existence results are obtained for the population model with or without volume filling.
|Original language||English (US)|
|Number of pages||39|
|State||Published - 2015|
Bibliographical noteKAUST Repository Item: Exported on 2021-10-15
Acknowledgements: The author acknowledges partial support from the Austrian Science Fund (FWF), grants P22108, P24304 and W1245 and the Austrian-French Program of the Austrian Exchange Service (OAD). Part of this manuscript was written during the stay of the author at the King Abdullah University of Science and Technology (KAUST) in Thuwal, Saudi-Arabia. The author thanks Peter Markowich for his kind invitation and support.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.